Find the perpendicular line of y = x - 9 that passes through the point (0, -6).

Steps to Solve

1. Identify the Slope of the Given Line

The given line is $y = x - 9$.

To find the slope, we can rewrite it in slope-intercept form, which is $y = mx + b$, where $m$ is the slope.

From the equation $y = 1x - 9$, we see that the slope $m$ is 1.

2. Find the Slope of the Perpendicular Line

The slope of a line that is perpendicular to another is the negative reciprocal of the original slope.

Since the original slope is $1$, the negative reciprocal is given by:

$$ m_{perpendicular} = -\frac{1}{1} = -1 $$

3. Use the Point-Slope Form to Find the New Line

Now we will use the point-slope form of the equation of a line, which is:

$$ y - y_1 = m(x - x_1) $$

Here, $(x_1, y_1)$ is the point through which the line passes, which is $(0, -6)$, and $m$ is the slope of the perpendicular line, which is $-1$.

Plugging in the values, we get:

$$ y - (-6) = -1(x - 0) $$

4. Simplify to Get the Equation of the Line

Now simplify the equation:

$$ y + 6 = -1x $$

Subtract $6$ from both sides:

$$ y = -x - 6 $$

This is the equation of the line that is perpendicular to the original line and passes through the point (0, -6).